Exercise 9.13. Let K be a field. 1. Show that a polynomial f(x) in K[x] of degree 2 either has a root in K or is irreducible in K [x]. 2. Show that a polynomial f(x) in K[x] of degree 3 either has a root in K or is irreducible.

Explain in detail
Show that any eigenvector of the matrix corresponding to lambda = -4 is a scalar multiple of the vector < 1, - 1,1 >..

Examine the following words and tell me how many permutations there are of the letters. We do not care about keeping track of the individual common letters. For example, in the word dad, there are two d 0 s and we want to treat the permutation d1d2a the same as.

Evaluate the limit:
1/b - 1/5/b - 5..

Exercise 1.20 (a) Let S Ax x E Rnt, where A is a given matrix. Show that S is a subspace of (b) Assume that S is a proper subspace of Rm. Show that there exists a matrix B such that S ty E R" I By 0). Hint: Use.

Estimate root 5 by finding the roots of f(x) = x^2-5. Use x0 =2 as your initial guess. Note that we can find this value without using the square root key on the calculator method by which many square root tables were created before the use of calculators. The TI-.

Exercise 1.2.8. Here are two important denitions related to a function f : A B. The function f
is one-to-one (1–1) if a1 I= a2 in A implies that f (a1) I= f (a2) in B. The function f is onto if,
given any.

Exercise 6.4.8 Let tri, r2, r3, be an enumeration of the set of rational numbers. For each rn E Q, define m(ar) f 1/2 for a rn for a S r Now, let h (a) un(a). Prove that h is a monotone function defined on all of R that is.

et A be a n x m-matrix over R and LA : Rn -> Rm the linear transformation defined by La(X) = A . X. Show that La is injective if and only if rank (A) = n. La is surjective if and only if rank(A) = m. La is.

Example: Let E21 = [1 0 0 -2 1 0 0 0 1]. (a) What kind of condition does a matrix B have to satisfy so that we can compute E21 B? (b) Describe in words what multiplying E21 to B, from the left, does to B?.

Expand (1+a)4.

Ex 4: Find the number at which each function is discontinuous. At which of these numbers is each function continuous from the right, from left or neither. x +1 if xsi 2x-x' b. f(x) f(x) if 1<x< 3="" <="" <x</x<>.

Evaluate the summation the lower limit is i= 0 and the upper limit is 99 using a greek sigma. ? (-1/2)^i. (Hint, there is 100 terms in the sum. Use the formula for the sum of terms of a geometric sequence. leave the answer with exponents rather than using a.